Why dividing by zero breaks arithmetic, and some calculators too
Ask an old mechanical calculator to divide by zero and it may churn forever. That is not a malfunction. It computes division by subtracting the divisor over and over until the dividend runs out, and subtracting zero never uses anything up, so the machine never reaches an answer.
The trouble shows up as soon as division is rewritten as multiplication. Six divided by three asks which number times three gives six, and the answer is two. Six divided by zero asks which number times zero gives six, and there is none, because anything times zero is zero. Zero divided by zero fails the other way: every number times zero gives zero, so no single value can be picked. Ordinary arithmetic therefore leaves the operation undefined, and allowing it lets clever-looking algebra reach absurd conclusions.
Everyday pictures show the same strain. Ten slices of bread make five two-slice sandwiches, but if a sandwich needs no bread at all, such as a lettuce wrap, you could make as many as you like. Sharing ten cookies between two friends gives each five; sharing them among zero friends is meaningless. A sugar-free cake with ten cups of flour and none of sugar is perfectly sensible, yet asking how many parts flour per part sugar has no numerical answer.
Geometry offers another view. A line's slope is its rise divided by its run, so a horizontal line has slope zero while a vertical line, with no run at all, has no defined slope and is sometimes described as infinitely steep. In calculus, as a denominator shrinks toward zero the fraction can grow without limit; if the top and bottom both head to zero, the result is an indeterminate form that depends on the exact functions involved.
Some number systems add a point at infinity, or signed infinities, to make the quotient mean something, though those points bring odd behaviour of their own. Computers vary: depending on the number type, the machine may answer with an infinity of either sign, flag the result as not a number, or halt with an error, and some theorem provers simply define the answer as zero.
Source: Division by zero
